• Title: Fitting Index

  • Series: Abstract Linear Algebra

  • Chapter: Some matrix decompositions

  • YouTube-Title: Abstract Linear Algebra 37 | Fitting Index

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  • Quiz Content

    Q1: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda = 0$ be an eigenvalue. What is the correct definition of the Fitting index?

    A1: The smallest $j \in \mathbb{N}$ such that $\dim(\mathrm{Ker}(A^j))$ $ = \dim(\mathrm{Ker}(A^{j+1}))$

    A2: $n$

    A3: $2$

    A4: $\dim(\mathrm{Ker}(A)) = 2$

    A5: The largest index $j \in \mathbb{N}$ such that $\dim(\mathrm{Ker}(A))$ $ = \dim(\mathrm{Ker}(A^{j}))$

    Q2: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda = 0$ be an eigenvalue. What is correct for range and kernel for every $j \in \mathbb{N}$?

    A1: $\dim( \mathrm{Ran}( A^j) ) $ $+ \dim ( \mathrm{Ker}( A^j) ) $ $= n$

    A2: $\dim( \mathrm{Ran}( A^j) ) $ $+ \dim ( \mathrm{Ker}( A^j) ) $ $= 0$

    A3: $\dim( \mathrm{Ran}( A ) ) $ $+ \dim ( \mathrm{Ker}( A^j) ) $ $= j$

    A4: $\dim( \mathrm{Ran}( A^n) ) $ $+ \dim ( \mathrm{Ker}( A^n) ) $ $= j$

    A5: $\dim( \mathrm{Ran}( A) ) $ $+ \dim ( \mathrm{Ker}( A) ) $ $= 1$

    Q3: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda = 0$ be an eigenvalue. The Fitting index is denoted by $d$. If we apply $A$ to the subspace $\mathrm{Ran}( A^d)$, what subspace do we get out?

    A1: $ \mathrm{Ran}( A^d )$

    A2: $ \mathrm{Ran}( A^{d-1} )$

    A3: $\mathbb{C}^n$

    A4: $\mathbb{C}^d$

    Q4: Let $A \in \mathbb{C}^{n \times n}$ for $n\geq 2$ and $\lambda = 0$ be an eigenvalue. What is not correct for the Fitting index $d$?

    A1: The smallest $j \in \mathbb{N}$ such that $\dim(\mathrm{Ker}(A^j)) $ $= \dim(\mathrm{Ker}(A^{j+1}))$

    A2: If $j> d$, we can have $\dim(\mathrm{Ker}(A^j)) $ $\neq \dim(\mathrm{Ker}(A^{j+1}))$

    A3: If $\dim(\mathrm{Ran}(A^j)) $ $\neq \dim(\mathrm{Ran}(A^{j+1}))$, then $j \leq d$.

  • Date of video: 2024-11-29

  • Last update: 2025-10

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